Improved and Generalized Algorithms for Burning a Planar Point Set

TL;DR

This paper introduces improved approximation algorithms for the point burning problem in planar point sets, providing PTAS in one dimension and better approximation ratios in two dimensions, along with complexity results.

Contribution

It presents the first PTAS for point burning in one dimension and improved approximation algorithms in two dimensions, along with complexity and generalization results.

Findings

PTAS achieved for point burning in one dimension.

Improved approximation ratios of (1.96296+ε) and (1.92188+ε) in two dimensions.

NP-hardness results for finding burning sequences even with given sources.

Abstract

Given a set $P$ of points in the plane, a point burning process is a discrete time process to burn all the points of $P$ where fires must be initiated at the given points. Specifically, the point burning process starts with a single burnt point from $P$, and at each subsequent step, burns all the points in the plane that are within one unit distance from the currently burnt points, as well as one other unburnt point of $P$ (if exists). The point burning number of $P$ is the smallest number of steps required to burn all the points of $P$. If we allow the fire to be initiated anywhere, then the burning process is called an anywhere burning process, and the corresponding burning number is called anywhere burning number. Computing the point and anywhere burning number is known to be NP-hard. In this paper we show that both these problems admit PTAS in one dimension. We then show that in two dimensions, point burning and anywhere burning are $(1.96296+\varepsilon)$ and $(1.92188+\varepsilon)$ approximable, respectively, for every $\varepsilon>0$, which improves the previously known $(2+\varepsilon)$ factor for these problems. We also observe that a known result on set cover problem can be leveraged to obtain a 2-approximation for burning the maximum number of points in a given number of steps. We show how the results generalize if we allow the points to have different fire spreading rates. Finally, we prove that even if the burning sources are given as input, finding a point burning sequence itself is NP-hard.